The authors establish that for algebraically integrable foliations on klt varieties, and for rank one foliations on threefolds, any two Mori fiber spaces are connected by Sarkisov links.
Brunella, Birational geometry of foliations, IMPA Monographs 1 (2015), Springer, Cham
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Sarkisov program for algebraically integrable and threefold foliations
The authors establish that for algebraically integrable foliations on klt varieties, and for rank one foliations on threefolds, any two Mori fiber spaces are connected by Sarkisov links.